Triplet loss

Background

Triplet loss trains embeddings so that an anchor is closer to a "positive" (same class) than to a "negative" (different class) by at least a margin. It is the workhorse of metric learning — face recognition, image retrieval, and contrastive representation learning all use it to shape an embedding space where distance means dissimilarity.

Problem statement

Implement triplet_loss(anchor, positive, negative, margin=1.0) using squared-Euclidean distances:

$$\mathcal{L} = \frac{1}{N}\sum_i \max\!\Big(0,\; \lVert a_i - p_i\rVert^2 - \lVert a_i - n_i\rVert^2 + \text{margin}\Big)$$

Input

• anchor, positive, negative — np.ndarray (N, D): the three embedding sets.
• margin — float: the required separation.

Output

Returns a float: the mean triplet loss over the batch ($\ge 0$).

Examples

Example 1

Input:  anchor = [[0, 0]], positive = [[2, 0]], negative = [[1, 0]], margin = 1.0
Output: 4.0

Explanation: $d_{ap} = 4$ and $d_{an} = 1$; the positive is farther than the negative, so the loss is $\max(0, 4 - 1 + 1) = 4$.

Constraints

• Use the squared Euclidean distance per embedding pair.
• The hinge $\max(0, \cdot)$ zeroes out triplets that already satisfy the margin.
• Average over the batch; the result is $\ge 0$.

Notes

• A loss of 0 means every triplet already respects the margin (anchor closer to its positive than its negative by at least margin) — the "easy" triplets that contribute no gradient.
• Mining hard / semi-hard triplets (those with positive loss) is what makes training efficient, since most random triplets are already easy.